Use the standardizing of normal random variables and show your work.
Length of metal strips produced by a machine are normally distributed with mean length of 150 cm and a standard deviation of 10 cm. Find the probability that the length of a randomly selected strip is
i/ Shorter than 165 cm?
ii/ Longer than 170 cm?
iii/ Between 145 cm and 155 cm?
Use the standardizing of normal random variables and show your work.
Since the mean is 150, the probability that the length is less than 150 is 50%. And the probability that the length is less than 165 is greater than that, i.e., it should be greater than 50%. What you found in i) is P(X > 165) = 1 - P(X < 165). The answer to ii) is correct. The answer to iii) is clearly wrong because it is quite likely that the length is between 145 and 155. P(145 < X < 155) = P(X < 155) - P(X < 145). You can also use another table here which lists the values for P(0 < Z < x). (It is easy to see that P(0 < Z < x) = 0.5 + P(Z < x).) This table is useful because P(145 < X < 155) = P(-0.5 < Z < 0.5) = 2P(0 < Z < 0.5).